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There is a nice survey on fairly elementary methods here by J. L. Nicolas, in a book called "Ramanujan Revisited." $$ $$ Meanwhile, there is an unconditional result which has not been mentioned, for $N \geq 3$ we have $$ \sigma(N) < e^\gamma \; N \log \log N + \frac{ 0.6482 N}{\log \log N} $$ I hope I am reporting this correctly, it is from a secondary source, attribution is to G.Robin, Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann, J. Math. Pures Appl. (9) 63 (1984) 187-213. $$ $$ The overall methodology is to consider the colossally abundant numbers of Alaoglu and Erdos (1944), which were eventually discovered to have also been present in the original version of Ramanujan's paper Highly Composite Numbers (1915). There is some history about why that section was omitted, I forget.